Finite Element Preconditioning on Spectral Element Discretizations for Coupled Elliptic Equations

<table>The condition numbers of the matrix <svg height="19.5625" id="M277" style="vertical-align:-3.35245pt" version="1.1" viewbox="0 0 27.174999 19.5625" width="27.174999" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.125,11.5)"><path d="M658 557l-9 -25q-129 23 -320 70q-210 -50 -320 -70l-9 25q232 77 318 116h22q89 -40 318 -116z" id="xF5C02"></path></g><g transform="matrix(.017,-0,0,-.017,.062,15.325)"><path d="M346 674l289 -674h-96l-82 200h-249l-81 -200h-96l289 674h26zM239 274h188l-94 229z" id="x1D5A0"></path></g>
<g transform="matrix(.012,-0,0,-.012,11.388,19.412)"><path d="M857 650l-6 -28q-44 -4 -61.5 -16.5t-29.5 -48.5q-11 -32 -37 -166l-78 -399h-29l-351 537h-4l-56 -276q-24 -120 -24 -164q0 -35 17.5 -46t75.5 -15l-6 -28h-245l7 28q41 2 62 14t31 44q10 30 41 171l53 245q8 44 6.5 60.5t-14.5 33.5q-10 15 -27 19.5t-64 6.5l6 28h153
l350 -516h5l48 257q25 131 25 171q0 34 -17.5 45t-77.5 15l7 28h240z" id="x1D441"></path></g>
<g transform="matrix(.008,-0,0,-.008,21.788,15.6)"><path d="M412 140l28 -9q0 -2 -35 -131h-373v23q112 112 161 170q59 70 92 127t33 115q0 63 -31 98t-86 35q-75 0 -137 -93l-22 20l57 81q55 59 135 59q69 0 118.5 -46.5t49.5 -122.5q0 -62 -29.5 -114t-102.5 -130l-141 -149h186q42 0 58.5 10.5t38.5 56.5z" id="x32"></path></g>
</svg> when <svg height="19.1" id="M278" style="vertical-align:-2.22495pt" version="1.1" viewbox="0 0 72.300003 19.1" width="72.300003" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,16.262)"><path d="M17 676h194v-259q53 56 122 56q82 0 135 -65t53 -169q0 -113 -63 -183t-163 -70q-71 0 -132 57l-79 -56h-12v607q0 30 -10 40.5t-45 17.5v24zM211 360v-263q0 -36 19 -57.5t50 -21.5q94 0 94 203q0 198 -93 198q-47 0 -70 -59z" id="x1D41B"></path></g><g transform="matrix(.017,-0,0,-.017,14.222,16.262)"><path d="M535 323h-483v50h483v-50zM535 138h-483v50h483v-50z" id="x3D"></path></g><g transform="matrix(.017,-0,0,-.017,28.926,16.262)"><path d="M290 -163h-170v866h170v-28q-79 -7 -94 -19.5t-15 -72.5v-627q0 -59 14.5 -71.5t94.5 -19.5v-28z" id="x5B"></path></g><g transform="matrix(.017,-0,0,-.017,34.791,16.262)"><path d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" id="x31"></path></g><g transform="matrix(.017,-0,0,-.017,42.95,16.262)"><path d="M95 130q31 0 61 -30t30 -78q0 -53 -38 -87.5t-93 -51.5l-11 29q77 31 77 85q0 26 -17.5 43t-44.5 24q-4 0 -8.5 6.5t-4.5 17.5q0 18 15 30t34 12z" id="x2C"></path></g><g transform="matrix(.017,-0,0,-.017,49.665,16.262)"><use xlink:href="#x31"></use></g><g transform="matrix(.017,-0,0,-.017,57.824,16.262)"><path d="M226 -163h-170v27q79 7 94 20t15 73v627q0 59 -15 72t-94 20v27h170v-866z" id="x5D"></path></g>
<g transform="matrix(.012,-0,0,-.012,63.688,8.1)"><path d="M649 676l-22 -187l-33 -2q3 56 -12 94q-8 20 -31.5 26.5t-86.5 6.5h-74l-90 -491q-4 -23 -6 -36.5t1 -25t7 -16.5t18 -9t27 -5t41 -3l-6 -28h-286l4 28q68 5 84 18.5t28 76.5l94 491h-55q-74 0 -100.5 -6t-41.5 -23q-24 -29 -54 -98l-32 1q32 98 53 188h22q7 -18 15 -22
t37 -4h417q23 0 33.5 5t25.5 21h23z" id="x1D447"></path></g>
</svg>.</table>

Journal of Applied Mathematics

fig1

Figure 1

Figure 1: Finite Element Preconditioning on Spectral Element Discretizations for Coupled Elliptic Equations 